Focal epilepsy — containment versus secondary generalisation (the first application of the spatial-localisation layer)

Chapter 43's spatial map gets its first application. Driving each region as a focal seizure focus, the foci partition into contained (the seizure stays focal) and broadcast (it secondarily generalises), and this partition is fixed by focus location, not ictal intensity. But broadcast is not the same as global hypersynchrony — two decoupled axes. Structural quantities, never felt experience.

The §43 spatial-localisation layer built a fixed spatial map of the frozen ephaptic kernel and handed it forward with one instruction: read it region by region. This chapter takes the map’s first reading, on the disorder it was built to answer. §25 certified the seizure as the network crossing the over-synchronisation threshold on the global order parameter R — but §25 drove the brain globally, one scalar coupling K_glob raised for the whole network at once, so it could not ask the single most consequential clinical question about a focal epilepsy: does a seizure that begins at one focus stay focal (a simple or complex partial seizure) or secondarily generalise — recruit circuits beyond the focus and spread to the whole brain? That question is intrinsically spatial, and §43 is exactly the layer that makes “where” representable. This chapter imports the SpatialField (it does not re-derive the kernel or the coupling map) and models a seizure focus as a strong focal excitatory ictal drive at one region, the rest at baseline, through the same k = κ/(1−|b|) map as §25 and §43 (no new constant), with the ictal intensity swept over {0.3, 0.5, 0.7, 0.9} and every sign required to hold at every intensity. Four results hold. F1 — the containment/broadcast partition is drive-invariant: each focus is contained (self-localising — the ictal change concentrates at the focus, the seizure stays focal) or broadcast (relay — the change lands harder off-target, the seizure secondarily generalises), and this binary partition is identical across the entire ictal-intensity sweep — the broadcast set {hippocampus, midbrain} fixed at every intensity, the other ten foci contained at every intensity. Whether a focal seizure stays focal or secondarily generalises is a fixed property of focus location, not of how intense the ictal drive is — the E1.3 self/relay class read on an ictal drive. F2 — broadcast is off-target dominance: for a broadcast focus the ictal local-coherence change lands harder on distal circuits than on the focus itself (mean off-target |Δc| > own |Δc|, ratio above 1 — the hippocampus at roughly 4.2×, the midbrain 1.6×) — the structural signature of a seizure recruiting circuits beyond its focus — while a contained focus concentrates at the focus (ratio below 1); the equivalence holds at every intensity. F3 — off-target spread is not global hypersynchrony (the honest no-tuning result): a clean, attractive hypothesis — “the broadcast, secondarily-generalising foci are exactly the foci that drive the whole brain into the §25 over-synchronisation state” — is false. The single largest global-reach focus is the cerebellum, which is contained, at every intensity, so the biggest global recruiter is not a broadcast focus; and the two broadcast foci move global synchrony in opposite directions — the midbrain raises R toward the over-sync state, the hippocampus lowers it — so “broadcast” carries no consistent global-synchrony sign at all. Off-target spread and global hypersynchrony are two distinct, decoupled spatial axes: secondary-generalisation propagation is site-determined, not reducible to a single “spread = hypersynchrony” rule. The refuted hypothesis is reported honestly — the E1.4 lesson made concrete for epilepsy. F4 — the broadcast set is a coherent minority relay-hub class: {hippocampus, midbrain} is simultaneously the E1.3 relay set, the off-target-dominant set (F2), a strict minority (2 of 12 — containment is the structural default), and disjoint from the global-reach hub (cerebellum, F3) — one coherent class of limbic/brainstem relay hubs. Secondary generalisation is structurally the exception, carried by specific relay hubs, not a global property of the ictal drive. A direction-only [L] correspondence is noted, never a magnitude or a prediction: clinically most focal seizures remain focal, and mesial-temporal (hippocampal) foci are the paradigmatic secondarily-generalising epilepsy — consistent with the hippocampus sitting in the broadcast set. A uniform (zero) ictal drive reproduces the frozen M9 coordination anchor (R = 0.38961455156) bit-for-bit and the off-state field equals the baseline exactly (S5) — a pure structural read on the frozen kernel (engine 0fbf4988…, byte-unchanged; SpatialField reused, not re-derived; no new tuned constant). The firewall is absolute: a containment/broadcast class is a structural spatial quantity of the coupling model, never the felt locus of a seizure, and not a real electrode, an EEG/SEEG localisation, a current density, a seizure-propagation map, a prediction of which patient’s seizures will generalise, or surgical/resection guidance (Axis-A — consciousness_claim = 0, the hard problem stays open). Every magnitude is [O]; efficacy = 0; not medical advice.

The question §25 could not ask — does a focal seizure stay focal?

The over-synchronisation chapter established the seizure with a single, sharp claim: a seizure is the network crossing a threshold on the global order parameter, the whole connectome locking into a coherence higher than the healthy operating point. That claim is right, and it is the foundation this chapter builds on — but it was made with a scalar drive, the same coupling applied to every region at once. With only a global dial there is one kind of seizure: the whole brain over-synchronising. Real epilepsy is not like that. The great majority of epilepsies are focal — the seizure begins in one region, an epileptogenic focus — and the clinically decisive fact about a focal seizure is what happens next: it may stay confined to its focus and its immediate network (a focal seizure, with or without impaired awareness), or it may secondarily generalise, recruiting circuits far beyond the focus until it involves the whole brain. The difference is the difference between a brief, localised event and a generalised tonic-clonic seizure; it is one of the things a clinician most wants to anticipate.

That question cannot even be posed with a global scalar, because it is a question about space: it asks where a drive that starts in one place ends up. A scalar has no “one place”. The spatial-localisation layer removed exactly that limitation: it generalised the scalar to a per-node drive and certified, as its headline (E1.3), that each region’s focal footprint is either self-localising (the drive concentrates at the site) or a relay (the drive lands harder off-target), and that this classification is drive-invariant — the relay set {hippocampus, midbrain} fixed at every amplitude. The spatial chapter named focal-epilepsy containment-versus-spread as the first application that map would unlock, and left it owed. This chapter cashes it: it reads the self/relay map as a containment / secondary-generalisation map for focal seizures, and then asks the harder question the map makes possible — whether “broadcast” (spreading off the focus) is the same thing as the §25 global over-synchronisation, or a different property entirely.

The focal model — a seizure focus as a focal ictal drive on the spatial layer

The model is the smallest one that makes the question answerable, and it reuses the spatial layer wholesale. A focal seizure focus is a strong, localised excitatory drive: region i is driven at an ictal intensity b₀ while every other region sits at baseline. The drive uses the same effective-coupling map as §25 and the spatial layer — k(b) = κ/(1−|b|) for an excitatory bias — so an ictal focus is simply a region pushed hard up the coupling curve, with no new constant introduced. The whole apparatus is the imported SpatialField object: its frozen ∼1/r³ row-stochastic kernel, its baseline order parameter and coherence field, and its focal(), reach() and footprint_class() readers. This chapter derives no mechanism of its own; it shapes one drive profile — a single hot focus — and reads the consequences the layer already knows how to report.

Two readings answer the two halves of the clinical question. The first is the footprint class: drive a focus and compare its own local-coherence change to the mean change it induces off-target. If the focus’s own change dominates, the drive is contained — the seizure stays focal. If the off-target change dominates, the drive is broadcast — the seizure secondarily generalises. The second is the global reach |R − R₀|: how much driving that one focus moves the network-wide order parameter, which is the quantity §25 made the seizure out of. Containment-versus-broadcast is a local, off-target reading; global reach is the §25 over-synchronisation reading. Holding both lets the chapter test whether they are the same axis — and they are not. The ictal intensity is swept, not chosen: every sign reported below is required to hold at b₀ = 0.3, 0.5, 0.7 and 0.9 alike, so no number is fitted to make a result come out.

F1 — the containment/broadcast partition is drive-invariant

The first result is the spatial discriminant for focal epilepsy, and it is inherited intact from the layer. Driving each of the 12 regions in turn as a focal ictal focus, the foci partition cleanly into two classes. Ten foci are contained: the ictal change concentrates at the driven focus, more than it changes anywhere else — structurally, the seizure stays focal. Two foci are broadcast: driving them changes coordination off the focus more than at the focus itself — structurally, the seizure secondarily generalises, recruiting distal circuits. The broadcast set is {hippocampus, midbrain}.

The certification is that this partition does not move with ictal intensity. From the gentlest swept drive to the strongest, the same ten foci stay contained and the same two broadcast: the broadcast set {hippocampus, midbrain} is fixed at every intensity tested. Whether a focal seizure contains itself or broadcasts is therefore a fixed property of where the focus sits in the field — a drive-invariant connectome property, not a function of how hard the focus fires. This is the direct epileptological reading of the layer’s headline E1.3 result: the self/relay class is the containment / secondary-generalisation class, and it is set by focus location. It identifies, before any magnitude is quoted, which foci would tend to stay focal and which would tend to generalise — a structural statement about focus behaviour, grounded entirely in network position.

F2 — broadcast is off-target dominance: the structure of secondary generalisation

The second result turns the binary class of F1 into a legible quantity, and ties “broadcast” to the clinical meaning of secondary generalisation. For each driven focus, take the ratio of the mean off-target coherence change to the focus’s own change. For a broadcast focus this ratio is above 1: the ictal drive lands harder on distal circuits than on the focus itself — the hippocampus at roughly 4.2×, the midbrain at 1.6×. That is precisely the structural picture of a seizure that does not stay put: the focus fires, but the largest coordination change is elsewhere, in the circuits the focus recruits. For a contained focus the ratio is below 1: the change is concentrated at the focus, the distal circuits barely move — the seizure stays local.

The equivalence — a focus is broadcast if and only if its off-target change exceeds its own — holds at every swept ictal intensity. This is what makes the F1 partition more than a label: secondary generalisation is not a separate phenomenon bolted on, it is the same footprint read quantitatively — the off-target-to-own ratio crossing 1. It also makes plain what kind of claim is and is not being made. The ratio is a structural measure of where a coordination change lands on the frozen kernel; it is not a current, a recruitment rate, or a clinical spread velocity. The magnitude of any real secondary generalisation is [O]; what is asserted is the sign — that broadcast foci dominate off-target and contained foci dominate at the focus, robustly across intensity.

F3 — off-target spread is not global hypersynchrony: two distinct axes

The third result is the one the no-tuning discipline exists to surface: a clean hypothesis, tested, and reported as false rather than quietly dropped. The hypothesis is the natural one to reach for after §25. §25 made the seizure the global over-synchronisation — the whole network locking above the healthy operating point — so the obvious guess is that the foci that broadcast (secondarily generalise) are exactly the foci that drive the whole brain into that over-sync state: the foci with the largest global reach |R − R₀|. If that held, “spreads off the focus” and “drives global hypersynchrony” would be the same axis, and the chapter would reduce to a single number.

It does not hold, in two separate ways, both robust across the intensity sweep. First, the broadcast set is not the high-global-reach set. The single largest global-reach focus — the focus whose drive moves the network-wide order parameter the most — is the cerebellum, and the cerebellum is contained, at every swept intensity. The biggest global recruiter is a focus whose own footprint stays local; conversely the two broadcast foci sit in the middle of the reach ranking, not the top. “Recruits the global state the most” and “spreads off its own focus” are simply different properties of different regions. Second, the two broadcast foci do not even agree on the sign of their effect on global synchrony. Driving the midbrain focally raises the global order parameter — pushing the network toward the §25 over-sync state. Driving the hippocampus focally lowers it — pushing the network away from over-sync, toward desynchronisation. Both are broadcast foci, both secondarily generalise in the off-target sense, yet one drives global hypersynchrony and the other drives global desynchronisation; this opposite-sign split holds at every swept intensity. “Broadcast” carries no consistent global-synchrony sign at all.

The conclusion is the honest, deflationary one. Off-target spread (the F1/F2 broadcast property) and global hypersynchrony (the §25 over-sync axis, read here as global reach and its sign) are two distinct, decoupled spatial axes. A focus can broadcast to distal circuits without driving global over-sync (the hippocampus), and a focus can move the global state strongly while keeping its own footprint local (the cerebellum). Secondary-generalisation propagation is therefore site-determined — a property of which focus, decoupled from the global over-sync magnitude — and it cannot be collapsed into a single “spread = hypersynchrony” rule. This is the E1.4 finding made concrete for epilepsy: there is no universal law turning a focal drive into a global outcome; the map must be read region by region. The tidy single-axis story the physics refuses to support is not forced — the refuted hypothesis is the finding, and reporting it is the no-tuning discipline working as intended.

F4 — the broadcast set is a coherent minority relay-hub class

The fourth result steps back and asks what kind of thing the broadcast set is, and the answer is a single coherent structural class. The two broadcast foci {hippocampus, midbrain} are, at once, four things. They are the E1.3 relay set (their footprints land off-target). They are the off-target-dominant set of F2 (their off-to-own ratio exceeds 1). They are a strict minority — 2 of 12 — so containment is the structural default: the typical focus stays focal, and secondary generalisation is the exception, not the rule. And they are disjoint from the global-reach hub of F3 (the cerebellum), confirming that the broadcast class is its own property, not a relabelling of “drives the global state”. These four descriptions pick out the same two regions, across the whole intensity sweep — one coherent class of limbic and brainstem relay hubs. Structurally, secondary generalisation is the behaviour of a small, specific set of relay hubs, not a global feature of how hard a focus is driven.

There is a direction-only correspondence to clinical epilepsy worth naming — carefully, and graded [L], because it is a cited resemblance and not a derived or predicted quantity. Clinically, most focal seizures do remain focal, and the mesial-temporal epilepsies — hippocampal sclerosis the commonest — are the paradigmatic secondarily-generalising focal epilepsy, with the hippocampal/limbic network a recognised route of propagation. That the model places the hippocampus in the broadcast set, and keeps containment as the majority class, is consistent with that picture — a coherence of direction, nothing more. It is not a claim that this model predicts which individual patient’s seizures will generalise, nor which focus in a real brain broadcasts; those are clinical determinations made from real data. The model offers a structural rationale for why containment is the norm and broadcast the exception, and why limbic/brainstem hubs are the structural candidates for spread — a hypothesis about mechanism, gated for reproducibility, awaiting external test.

S5 — the engine-invariance guard: a zero ictal drive recovers the frozen anchor

As with every module in the series, the focal model is certified to be a pure structural read that adds nothing to the engine. When the ictal drive is set to zero everywhere — no focus driven, the spatial drive collapsed to nothing — the integration reproduces the frozen M9 coordination anchor R = 0.38961455156044245 bit-for-bit, identical to the engine’s own direct value at full precision, and the per-node local-coherence field equals the stored baseline field exactly. Turning the focus off recovers the frozen engine with nothing left over. The SpatialField is imported and the engine emerged read-only, confirmed byte-unchanged against the frozen tree hash (0fbf4988fc83…) with the M0–16 subtree identical. There is no new mechanism, no new measurement, and no new tuned constant: this chapter shapes one drive profile on a frozen kernel and reads four signs off it, all of which survive the ictal-intensity sweep.

The four results side by side — what the chapter establishes

The chapter is four certified statements about one object — a focal seizure focus on the spatial kernel. Read across a row to see one result and the discriminant that carries it; read down to see the chapter move from the containment/spread partition (the discriminant), through its quantitative content, to the honest decoupling from global over-sync, and finally to the structural class the broadcast foci form.

The table makes the chapter’s logic legible. F1 is the discriminant — a focus stays focal or generalises by its position. F2 is the quantitative content — broadcast means the change lands off-target. F3 is the honest boundary — broadcast is not the §25 over-sync axis, so the two cannot be conflated. F4 is the structural class — broadcast is the behaviour of a small set of relay hubs, containment the default. Together they are the first region-specific reading of the spatial map, on the disorder the map was built to answer.

What the chapter does not claim — the firewall

This is a chapter about epilepsy, and the boundary of what it asserts must be stated without hedging. Every quantity certified here is a structural spatial quantity — a containment/broadcast class, an off-target-to-own ratio, a global reach, all of them properties of how a coupling model coordinates across a fixed kernel — and none of them is a claim about the felt quality of a seizure. That a focus “broadcasts” is a statement about where coordination changes in the connectome model; it is not a statement about what a seizure feels like, about an aura, or about awareness, and it is certainly not a claim that experience is located at the focus. This is the Axis-A firewall, held exactly as in every chapter of the series: consciousness_claim = 0, and the hard problem of experience stays open. Giving a seizure focus a spatial address in the model does not give the experience of a seizure one.

The boundary to clinical reality is firmer still, and it matters more here than almost anywhere in the series. The per-node bias is not a real ictal drive, the local-coherence field is not a real current density or an EEG/SEEG recording, the footprint is not a real seizure-propagation map, and the containment/broadcast class is not a prediction of which patient’s seizures will secondarily generalise. Real seizure propagation is heterogeneous — it runs on white-matter tractography, the individual epileptogenic network, ictal recruitment dynamics, and the patient’s own connectome and pathology — and this module asserts only the sign and structure of a focal ictal drive on the frozen kernel, not that any real focal seizure follows it. Epilepsy diagnosis, seizure localisation, and the decision to resect a focus are external clinical determinations, made by clinicians with real data; nothing here is localisation, prognosis, or surgical guidance, and the [L] correspondence in F4 is a cited resemblance of direction, never a patient-level claim. Every magnitude is [O]: representative reads over a swept ictal intensity, never quantities fitted to a target. There is no new mechanism, no new measurement beyond the read-only field, and no new tuned constant in this chapter; the SpatialField is imported and the engine is byte-unchanged. Nothing here is a cure, a treatment, a diagnosis, a localisation, a prognosis, or a recommendation. efficacy = 0; this is not medical advice. What the chapter offers is one structural thing: whether a focal seizure tends to stay focal or secondarily generalise is, in this model, a fixed property of focus position — and that property is not the same as driving the whole brain into hypersynchrony, so the spatial story of focal epilepsy, like every disease on this layer, must be told region by region.